# Linear Algebra

## Contents

## Theorems

solution Find the eigenvalues of the matrix

solution Define the adjoint of a matrix.

solution Define a self-adjoint matrix.

solution Define a unitary matrix.

solution Show that

solution Show that

solution Show that is self-adjoint.

solution Show that the identity matrix is self-adjoint.

solution Show that the zero matrix is self-adjoint.

solution Show that

solution Let be an matrix such that , and let be the identity matrix. Prove that .

solution Let be an matrix. Prove that .

## Matrices

### Basic Problems

solution If and evaluate

solution Find such that .

solution If Show that

Solution If w is cube root of unity,show that

solution If for all integral values of n,show that

solution Find the value of determinant of the matrix

solution Show that

solution Prove that the determinant of the matrix

solution Show that

solution Without expanding the determinant of the matrix prove that

solution Prove that

solution If a,b,c are distinct, and then prove that

solution Show that

solution Prove that

solution Without expanding the determinant prove that

solution Solve for x given

solution Show that the determinant of the matrix is independent of theta.

solution Show that

### Inverse & Rank of a Matrix

solution If A and B are two non-singular matrices of the same type,then adjoint(AB)=(adjoint B)(adjoint A)

solution If A,B are invertible matrices of the same order,then

Solution Compute the adjoint of the matrix

solution Find the adjoint and inverse of

solution Determine the rank of

solutionFind the rank of A,rank of B ,

solution Determine the values of b such that the rank of the matrix A is 3.

solution Find the non-singular matrices P and Q such that the normal form of A is PAQ where . Hence find its rank.

solution Find P and Q such that the normal form of is PAQ. Hence the find the rank.

solution A=, B=. Find the rank of and .

solution Solve by Cramer's rule

## Inner Products

solution Define an inner product.

solution Show that

solution Show that

solution Show that

solution Show that

solution Show that

solution Show that

solution Show that

solution Show that

solution Show that

solution Show that is always real.

## Vector Algebra

### Vector Addition

solution If are the medians of a triangle,then prove that

solution If G is the centroid of the triangle ABC,prove that where are the vertices of the triangle ABC and is the point vector

solution The position vectors of A and B are respectively.Find the position vector of the point which divides the line segment AB in the ration 2:3.

solution If then express as a linear combination of and .

solution If ,then find the position vector of D.

solution If is the position vector whose point is .Find the coordinates of a point B such that ,the coordinates of A are

solution Find a vector of magnitude 6units which is parallel to the vector

solution Find the magnitude of the vector

solution If the position vectors of A and B are respectively,find the unit vector in the direction of AB.

solution If the position vectors of A and B are respectively,determine the direction cosines of

solution In a triangle ABC if and D is the mid point of the side BC, then find the length of AD.

solution Show that the points represented by are collinear.

solution Show that the points A,B,C,D with position vectors are not coplanar.

solution Prove that three points whose vectors are form an equilateral triangle.

solution Show that the triangle ABC whose vertices are is isoscles and right angled.

solution Obtain the point of intersection of the line joining the points with the plane through the points and

### Vector Product

solution Show that the points whose position vectors are are the vertices of a right angled triangle.

solution Find a vector which is perpendicular to both and where

solution If are mutually perpendicular vectors of equal magnitude,show that is equally inclined to

solution Dot products of the vectors are respectively.Find the vector.

solution Find the vector equation of a plane which is at a distance of 5units from the origin and which has as a normal vector.

solution Find the vector equation of the plane through the point and perpendicular to the vector .

solution Find the equation of the plane passing through the point and parallel to the plane

solution If ,then write

solution Determine the unit vector perpendicular to both the vectors

solution Find the vector area of a parallelogram whose diagonals are determined by the vectors

solution Find a vector of magnitude 3 and which is perpendicular to both the vectors

solution IF are the vertices of a triangle, find its area.

solution Find a unit vector perpendiculars to the plane ABC where

solution Let .If a vector satisfies and then find the vector

solution Find the volume of the parallelopiped whose edges are .

solution Show that the four points having position vectors are not coplanar.

solution If the two vectors are two vectors,find the projection of on

solution Reduce the equation to normal form and hence find the length of the perpendicular from the origin to the plane.

solution Find the angle between the planes

solution Find the value of lambda for which the four points with position vectors are coplanar.

solution Find the volume of the tetrahedron with vertices

solution Find the vector equation of the line passing through three non-collinear points .Also find its cartesian equation.

solution Find the equation of the plane passing through the points and parallel to

solution If and the vectors are non-coplanar,then prove that

solution Find the perpendicular distance from the origin to the plane passing through the points

### Jordan